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Theorem xmstopn 23036
Description: The topology component of an extended metric space coincides with the topology generated by the metric component. (Contributed by Mario Carneiro, 26-Aug-2015.)
Hypotheses
Ref Expression
isms.j 𝐽 = (TopOpen‘𝐾)
isms.x 𝑋 = (Base‘𝐾)
isms.d 𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋))
Assertion
Ref Expression
xmstopn (𝐾 ∈ ∞MetSp → 𝐽 = (MetOpen‘𝐷))

Proof of Theorem xmstopn
StepHypRef Expression
1 isms.j . . 3 𝐽 = (TopOpen‘𝐾)
2 isms.x . . 3 𝑋 = (Base‘𝐾)
3 isms.d . . 3 𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋))
41, 2, 3isxms 23032 . 2 (𝐾 ∈ ∞MetSp ↔ (𝐾 ∈ TopSp ∧ 𝐽 = (MetOpen‘𝐷)))
54simprbi 500 1 (𝐾 ∈ ∞MetSp → 𝐽 = (MetOpen‘𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115   × cxp 5526  cres 5530  cfv 6328  Basecbs 16461  distcds 16552  TopOpenctopn 16673  MetOpencmopn 20510  TopSpctps 21515  ∞MetSpcxms 22902
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-rab 3135  df-v 3473  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-opab 5102  df-xp 5534  df-res 5540  df-iota 6287  df-fv 6336  df-xms 22905
This theorem is referenced by:  imasf1oxms  23074  ressxms  23110  prdsxmslem2  23114  tmsxpsmopn  23122  xmsusp  23154  cmetcusp1  23935  minveclem4a  24012  minveclem4  24014  qqhcn  31239  rrhcn  31245  rrexthaus  31255  dya2icoseg2  31543  sitmcl  31616
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